A subgroup H of a group G is said to be pronormal in G if each of its conjugates $H^g$ in G is already conjugate to it in the subgroup $\langle H,H^g\rangle $. The aim of this paper is to classify those (locally) finite simple groups which have only nilpotent or pronormal subgroups.

A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

The pronorm of a group G is the set $P(G)$ of all elements $g\in G$ such that X and $X^g$ are conjugate in ${\langle {X,X^g}\rangle }$ for every subgroup X of G. In general the pronorm is not a subgroup, but we give evidence of some classes of groups in which this property holds. We also investigate the structure of a generalised soluble group G whose pronorm contains a subgroup of finite index.

We extend to soluble $\text{FC}^{\ast }$-groups, the class of generalised FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J. 28(3) (2002), 241–254], the characterisation of finite soluble T-groups obtained recently in Kaplan [‘On T-groups, supersolvable groups, and maximal subgroups’, Arch. Math. (Basel) 96(1) (2011), 19–25].